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Mirrors > Home > MPE Home > Th. List > Mathboxes > submateqlem2 | Structured version Visualization version GIF version |
Description: Lemma for submateq 33463. (Contributed by Thierry Arnoux, 26-Aug-2020.) |
Ref | Expression |
---|---|
submateqlem2.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
submateqlem2.k | ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) |
submateqlem2.m | ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) |
submateqlem2.1 | ⊢ (𝜑 → 𝑀 < 𝐾) |
Ref | Expression |
---|---|
submateqlem2 | ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fz1ssnn 13559 | . . . . . 6 ⊢ (1...(𝑁 − 1)) ⊆ ℕ | |
2 | submateqlem2.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) | |
3 | 1, 2 | sselid 3971 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
4 | 3 | nnge1d 12285 | . . . 4 ⊢ (𝜑 → 1 ≤ 𝑀) |
5 | submateqlem2.1 | . . . 4 ⊢ (𝜑 → 𝑀 < 𝐾) | |
6 | 4, 5 | jca 510 | . . 3 ⊢ (𝜑 → (1 ≤ 𝑀 ∧ 𝑀 < 𝐾)) |
7 | 2 | elfzelzd 13529 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
8 | 1zzd 12618 | . . . 4 ⊢ (𝜑 → 1 ∈ ℤ) | |
9 | submateqlem2.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) | |
10 | 9 | elfzelzd 13529 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
11 | elfzo 13661 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ∈ (1..^𝐾) ↔ (1 ≤ 𝑀 ∧ 𝑀 < 𝐾))) | |
12 | 7, 8, 10, 11 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ↔ (1 ≤ 𝑀 ∧ 𝑀 < 𝐾))) |
13 | 6, 12 | mpbird 256 | . 2 ⊢ (𝜑 → 𝑀 ∈ (1..^𝐾)) |
14 | 2 | orcd 871 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ (1...(𝑁 − 1)) ∨ 𝑀 = 𝑁)) |
15 | submateqlem2.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
16 | nnuz 12890 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
17 | 15, 16 | eleqtrdi 2835 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1)) |
18 | fzm1 13608 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘1) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 ∈ (1...(𝑁 − 1)) ∨ 𝑀 = 𝑁))) | |
19 | 17, 18 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ (1...𝑁) ↔ (𝑀 ∈ (1...(𝑁 − 1)) ∨ 𝑀 = 𝑁))) |
20 | 14, 19 | mpbird 256 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) |
21 | 3 | nnred 12252 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
22 | 21, 5 | ltned 11375 | . . . 4 ⊢ (𝜑 → 𝑀 ≠ 𝐾) |
23 | nelsn 4665 | . . . 4 ⊢ (𝑀 ≠ 𝐾 → ¬ 𝑀 ∈ {𝐾}) | |
24 | 22, 23 | syl 17 | . . 3 ⊢ (𝜑 → ¬ 𝑀 ∈ {𝐾}) |
25 | 20, 24 | eldifd 3952 | . 2 ⊢ (𝜑 → 𝑀 ∈ ((1...𝑁) ∖ {𝐾})) |
26 | 13, 25 | jca 510 | 1 ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∖ cdif 3938 {csn 4625 class class class wbr 5144 ‘cfv 6543 (class class class)co 7413 1c1 11134 < clt 11273 ≤ cle 11274 − cmin 11469 ℕcn 12237 ℤcz 12583 ℤ≥cuz 12847 ...cfz 13511 ..^cfzo 13654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-fzo 13655 |
This theorem is referenced by: submateq 33463 |
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